Is there a Noether normalisation lemma for finitely generated (flat) algebras over $\mathbf{Z}$ (or more generally principal ideal domains)? It seems one can tensorise with the quotient field and then apply the usual Noether normalisation lemma. I couldn't find this in the literature, so I suspect it is wrong.

$\begingroup$ Related? mathoverflow.net/questions/42276/… $\endgroup$ – Asvin Oct 20 '17 at 17:14

1$\begingroup$ Isn't this the same question? math.stackexchange.com/questions/213336/… $\endgroup$ – Asvin Oct 20 '17 at 17:16

$\begingroup$ @Asvin: Thanks! Can one omit the localisation in the case of a PID and a flat algebra? $\endgroup$ – user19475 Oct 20 '17 at 17:33

$\begingroup$ I haven't actually gone through the links myself. I just remembered seeing similar questions before. Maybe you will find the formulation here more useful? mathoverflow.net/a/60716/58001 $\endgroup$ – Asvin Oct 20 '17 at 17:47

$\begingroup$ @TimoKeller No, just look at the counterexample from the second question, with $\mathbb Z[1/2]$. $\endgroup$ – Will Sawin Oct 20 '17 at 19:17
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you can consult the paper Corrigendum to “Noether Normalization theorem and dynamical Gröbner bases over Bezout domains of Krull dimension 1” [J. Algebra 492 (15) (2017) 5256] by Maroua Gamanda and Ihsen Yengui.
The link is https://www.sciencedirect.com/science/article/pii/S002186931930314X